Modelling the Federal Debt since Confederation
Overview
In this lesson, students will extract data on Canada’s federal debt from
Statistics Canada’s
E-STAT database. As a class, data on the federal debt from 1867 to
2007 will first be analyzed in total, and then modelled with a mathematical function. By
analyzing the data for separate time periods, students will apply function models and realize that
Canada’s federal debt is best modelled by a series of several different function types. Students
will be asked to provide evidence for their function choice for their assigned time period through
the use of difference tables and analysis of r2 values and residuals.
Contributor: Jennifer Hall, Statistics Canada
Objectives
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Collect secondary data that may be represented by linear, quadratic, and exponential
functions in a piecewise manner over time
Through investigation, determine the relationships between the graphs and the equations of
linear, quadratic, and exponential functions
Use difference tables to determine what equation type best represents data
Explain the role of individual variables in linear, quadratic, and exponential functions
Fit the equations of linear, quadratic, and exponential functions to a scatter plot using an
informal process, using technology
Research social, political, and economic reasons for the pattern that data follow
Suggested grade levels and subject areas
Grade 12
Mathematics (Advanced Functions)
Duration
Two to three 75 minute periods
Materials
Computers with Internet access and dynamic statistical software (e.g., Fathom). Models can
also be added to data in Excel.
Computer projector
E-STAT account
Student instructions
Student worksheet
Teacher worksheet
Graph for 1867 to 2007
Graphs for each time period
Prior knowledge
Linear functions in slope-intercept form: y = mx + b
Quadratic functions in vertex form: y = a (x – h)2 + k
Exponential functions in the form y = cax
Difference tables and residuals
Coefficient of determination, r2
Basic knowledge of E-STAT and dynamic statistical software (e.g., Fathom)
Classroom instructions
1) If necessary, review important properties of linear, quadratic, and exponential functions, as
well as the topics of difference tables, residuals, and the coefficient of determination.
2) Begin the lesson by playing the Statistics Canada Flash presentation ‘Time line’ for the
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class. This presentation highlights major events that occurred in Canadian history from
1901 to 2001. This Flash presentation and a text version of the information are located at:
http://www.statcan.gc.ca/kits-trousses/access-acces/edu06_0007-eng.htm.
Hold a brief class discussion on the topic of the federal debt in Canada to assess students’
prior knowledge and share information on the topic. Have students hypothesize how some
of the major events from the Flash presentation would impact the federal debt.
Display the graph of federal debt data from 1867 to 2007 from E-STAT using an overhead
projector. Ask students to determine by inspection what one function type they feel would
best approximate the data (Answer should be ‘exponential’).
Import the federal debt data into your dynamic statistical software program using the steps
provided in the student instructions. Demonstrate how to fit an exponential function to the
data, by altering the variable values. Discuss with the students how they would test which
function models best fit the curve (e.g., difference tables, residuals, r2 values).
Discuss with the students how the data could be better approximated by several
mathematical functions by partitioning the data into several time periods. Again, ask the
students to determine by inspection by which function types they feel that the data could be
approximated. (Answers may vary – e.g., linear until 1945, then exponential until the late
1990s, and then linear after that)
Divide the class into pairs or groups, and have them complete the worksheet for the
following 11 time periods:
a) 1867 to 1883 [Linear]
b) 1884 to 1908 [Linear]
c) 1909 to 1920 [Quadratic]
d) 1921 to 1930 [Linear]
e) 1931 to 1937 [Linear]
f) 1938 to 1946 [Exponential has R2=0.97. Quadratic has R2=0.998]
g) 1947 to 1974 [Quadratic]
h) 1975 to 1988 [Exponential and Quadratic have exactly the same R2]
i) 1989 to 1997 [Linear]
j) 1998 to 2007 [Linear] * Note that 2008 data will be available in E-STAT for the school
year starting in September 2009. It may change the function type that best models.
Listed in square brackets beside the time period is the type of function that best fits the data
for that time period. You may wish to divide the class into 10 groups (one per time period),
or assign multiple time periods to a group to ensure students are responsible for modelling
data with multiple function types. For the groups assigned to the first (1867 to 1883) and last
(1998 to 2007) time periods, in Question #9 of the worksheet, they will only make
comparisons with one other group.
6) Once each group has completed their analysis, you may wish to display all the graphs
together (e.g., tape them up on the chalkboard) to better show how the data break down into
several function types. Be sure that all students are using the same units for the value of the
federal debt (e.g., millions of dollars). Have the students discuss whether there are trends that
are common to multiple time periods, and discuss what historic events occurred during those
periods (e.g., World War I and II, the Great Depression).
Enrichment
Have students repeat this process for their province’s debt using table 385-0014 (Note: Under
‘Government sectors’, select ‘Provincial and territorial general government’. Under ‘Balance
sheet’, select ‘Net financial debt’.) Data are available from 1970 onwards.
Have students research economic factors that influence the federal debt, such as the role of the
Bank of Canada. The references below should get students started:
 Frequently asked questions about the Bank of Canada:
http://www.bankofcanada.ca/en/faq/faq_bank_monetary.html
 Report on the Canadian Debt-Strategy Model:
http://www.bankofcanada.ca/en/review/rev_summer2008.html
Challenge students to search on the E-STAT CANSIM database to find other time series data
(among the more than 38 million time series) that can be modelled by combination of functions.
They can import these data into a statistical software program and attempt to plot functions that
fit their data. Note: Several examples of individual and composite functions are provided at
www.statcan.gc.ca/edu/edu05_0018c-eng.htm. If your students find something interesting to
model, please send us a note at mdm4u@statcan.gc.ca. We’d certainly like to read it and may
add it to the bank of examples we use!
Evaluation
Students will be informally assessed on their work habits and computer skills throughout this
activity. They can be formally assessed via the worksheet, which can be marked using a
marking scheme of the teacher’s choice. A sample completed worksheet is provided for one
time period. Sample graphs are provided for each of the 10 time periods with function models
that approximate the data.
Graph of Canadian Federal Debt, 1867 to 2007
Source: Statistics Canada. Table 385-0010 - Federal government debt, for fiscal year ending March 31,
annual (dollars) (graph), CANSIM (database), Using E-STAT (distributor).
http://estat.statcan.ca/cgiwin/cnsmcgi.exe?Lang=E&ESTATFile=EStat\English\CII_1_E.htm&RootDir=ESTAT/
(accessed: August 21, 2008)
Student Instructions
Modelling Federal Debt since Confederation
In this activity, you will retrieve data from Statistics Canada’s E-STAT database on the federal
debt, and fit mathematical function models to individual time periods since Confederation.
Mathematical models can be useful in understanding past trends and making future predictions.
Name:
Time Period:
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E-STAT Instructions
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Go to the E-STAT website (http://estat.statcan.ca).
Select English.
Click on the Accept and enter button on the next page.
Click on Search CANSIM on the left sidebar.
Type 385-0010 in the Search box and click on the Search button to retrieve Table 385-0010
– Federal government debt, for fiscal year ending March 31, annual (dollars).
On the Subset selection page:
 Under Geography, select Canada
 Under Government debt, select Gross federal government debt
 Under From, select the earliest year in your assigned time period
 Under To, select the most recent year in your assigned time period
Click on the Retrieve as individual Time Series button.
On the Output specification page, select one of the following formats:
 Plain text: Table, time as rows
 HTML, table: Time as rows
 CSV: Time as rows.
The output selected will depend on your software. Ask your teacher which format works best
with your school’s software.
Click on the Retrieve now button.
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Software Instructions
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Import the data into your software program. Consult with your teacher or classmates if you
have problems importing the data.
Save your file as Federal Debt [Dates]. For example, if you have data from 1867 to 1883,
your file name would be: Federal Debt 1867 to 1883.
Create a new variable that represents the number of years since the first data point, using
the following formula:
Years since [First Data Point Year] = Year – [First Data Point Year].
Using the above example, the formula would be: Years since 1867 = Year - 1867
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Create a scatter plot with Years since [First Data Point Year] (e.g., Years since 1867) on
the x-axis and Federal debt (Millions) on the y-axis.
Determine what type of function best models the data in your scatter plot, using a variety of
methods (e.g., visual inspection, r2 values, residuals).
Plot that function type on your graph and alter the variable values until you have a close
approximation of the function to the data.
Student Worksheet
Modelling Federal Debt since Confederation
Name:
Time Period:
________________________________________________________________________
Questions
1)
What function type do you think best approximates your data?
2)
Why did you select this function type? Be sure to include difference tables and r 2 values in
your explanation.
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What is the basic form of the equation (e.g., y = mx + b) for the function you selected?
4)
What does each variable represent in the equation? (e.g., What do m and b represent in a
linear equation?)
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Specific to this dataset, what does each variable represent in the real world?
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What are your best values for each variable?
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What do the values of these variables mean in terms of the real world?
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Write your equation here, with your best values inserted.
9)
Consult with the students who analysed the data in the time periods immediately before and
after your assigned time period. What are the similarities and differences in their data and
your data, and in the functions you selected?
10) What historical events may have contributed to the shape of your graph, and how? Consult
the following resources to learn about historical events of the era:
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20th Century Timeline (as shown at the start of class):
http://www12.statcan.ca/english/census01/products/analytic/companion/age/timeline.cfm
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The Living Census (go to ‘Census Timeline’ and then select a census year that occurred
in your time period to learn about major events of the era)
http://thelivingcensus.statcan.ca/
11) Print a copy of your graph (with the function model and best values for your variables
showing) and hand it in with your answers to the above questions.
Student Worksheet
Modelling Federal Debt since Confederation
TEACHER VERSION
Name: Teacher
Time Period: 1989 to 1997
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Questions
1) What function type do you think best approximates your data?
Linear
2)
Why do you think this? Be sure to include difference tables and r2 values in your
explanation.
Year
1989
1990
1991
1992
1993
1994
1995
1996
1997
Debt (Millions)
379,993
406,606
444,557
476,104
514,357
557,604
595,877
634,939
651124
First Difference
26,613
37,951
31,547
38,253
43,247
38,273
39,062
16185
The r2 value of the least-squares line (line of best fit) is 1.00, which means the data are
approximated by a linear model very well. Although the first differences are not exactly
equal, this is to be expected because we are using real data. The first differences are close
though, and the data are best approximated by a linear model.
3)
What is the basic form of the equation (e.g., y = mx + b) for the function you selected?
y = mx + b
4)
What does each variable represent in the equation? (e.g., What do m and b represent in a
linear equation?)*
m represents the slope of the line.
b represents the y-intercept of the line.
5)
Specific to this dataset, what does each variable represent in the real world?
For this dataset, m represents the rate at which the federal debt is increasing or decreasing
annually whereas b represents the value of the federal debt for the first year of the data.
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What are your best values for each variable?
My best value for m is 36,000 and my best value for b is 374,000.
7)
What do the values of these variables mean in terms of the real world?
The m value of 36,000 means that the federal debt is increasing at a rate of $36,000,000 per
year, whereas the b value of 338,400 means that the value of the federal debt in the first year
of data is $374,000,000. (Note: Federal debt values are provided in thousands of dollars.)
8)
Write your equation here, with your best values inserted.
y = 36,000x + 374,000
9)
Consult with the students who analysed the data in the time periods immediately before
and after your assigned time period. What are the similarities and differences in their data
and your data, and in the functions you selected?
Previous time period: 1975 to 1988
In this time period, the debt was best modelled by a quadratic or an exponential function,
which means it changed more rapidly each year than during my time period. However, in
both their and my time period, the debt is increasing.
Next time period: 1998 to 2007
In this time period, debt decreased linearly, whereas in my period, debt increased linearly. In
the next time period, the data points do not follow the linear model as well as the data in my
time period. Also, the linear model in the next time period is not as steep as that in my time
period.
10)
What historical events may have contributed to the shape of your graph, and how?
Answers vary.
11)
Print a copy of your graph (with the function model and best values for your variables
showing) and hand it in with your answers to the above questions.
Graphs for each time period
a) 1867 to 1883 [Linear]
b) 1884 to 1908 [Linear]
c) 1909 to 1920 [Exponential]
d) 1921 to 1930 [Linear]
e) 1931 to 1937 [Linear]
f) 1938 to 1946 [Exponential]
g) 1947 to 1974 [Quadratic]
h) 1975 to 1988 [Exponential]
1975 to 1988 [Quadratic]
i) 1989 to 1997 [Linear]
j) 1998 to 2007 [Linear]